simple geometric model
1. description of all variables
- see parameter references from similar models for more detailed notes on parameter choices
| description | value | units | reference | type | python variable | associated equation | LaTeX variable |
|---|---|---|---|---|---|---|---|
| invagination height | \(nm\) | input (variable) | Hi | \(H_{i}\) | |||
| membrane tension | 0.001 | \(pN/nm\) | hassinger_design_2017 | fixed | sigma | \(\sigma\) | |
| total plane radius | 2000 | \(nm\) | fixed | At | \(A_{t}\) | ||
| final vesicle radius | 50 | \(nm\) | kaksonen_mechanisms_2018 | fixed | vesicleradius | \(R_{v}\) | |
| coat bending rigidity | 160 | \(pN nm\) | dmitrieff_membrane_2015 | fixed | K | \(K\) | |
| spontaneous curvature | 1/50 | \(nm^{-1}\) | hassinger_design_2017 | fixed | Ci | \(C_{i}\) | |
| droplet-cytosol surface tension | 0.005 | \(pN/nm\) | jawerth_salt-dependent_2018 | fixed | gammad | \(\gamma_{d}\) | |
| droplet-membrane wetting tension | 0.002 | \(pN/nm\) | fixed | gammam | \(\gamma_{m}\) | ||
| initial droplet plane radius | fixed | ||||||
| droplet-invagination surface tension | 0.002 | \(pN/nm\) | fixed | gammai | \(\gamma_{i}\) | ||
| turgor pressure | 0.1 | \(pN/nm^2\) | ma_mechanics_2019 | fixed | P | \(P\) | |
| relative invagination height | unitless | input (variable) | h | rel-h | \(h\) | ||
| droplet volume | \(nm^3\) | intermediate output | Vd | \(V_{d}\) | |||
| invagination surface area | \(nm^2\) | intermediate output | Si | invagination-area | \(S_{i}\) | ||
| initial droplet spherical radius | 600 | \(nm\) | intermediate output | dropradius | |||
| droplet-flat membrane contact area | \(nm^2\) | intermediate output | Sm | md-area | \(S_{m}\) | ||
| invagination spherical radius | \(nm\) | intermediate output | Ri | invagination-radius | \(R_{i}\) | ||
| droplet-cytosol contact area | \(nm^2\) | intermediate output | Sd | dc-contact-area | \(S_{d}\) | ||
| droplet contact angle | rad | intermediate output | thetad | droplet-contact-angle | \(\theta_{d}\) | ||
| invagination contact angle | rad | intermediate output | thetai | invagination-angle | \(\theta_{i}\) | ||
| invagination plane radius | \(nm\) | intermediate output | Ai | invagination-plane-radius | \(A_{i}\) | ||
| droplet + invagination plane radius | \(nm\) | intermediate output | Ad | droplet-plane-radius | \(A_{d}\) | ||
| outside membrane surface area | \(nm^2\) | intermediate output | So | outside-membrane-area | \(S_{o}\) | ||
| invagination volume | \(nm^3\) | intermediate output | Vi | invagination-volume | \(V_{i}\) | ||
| droplet + invagination volume | \(nm^3\) | intermediate output | Vb | di-volume | \(V_{b}\) | ||
| droplet + invagination height | \(nm\) | intermediate output | Hd | di-height | \(H_{d}\) | ||
| droplet + invagination spherical radius | \(nm\) | intermediate output | Rd | di-cap-radius | \(R_{d}\) | ||
| free energy | \(pN \cdot nm\) | output | F | f-forces f-shapes | \(F\) | ||
| membrane tension energy | \(pN \cdot nm\) | output | membranetension | ||||
| bending energy | \(pN \cdot nm\) | output | bendingenergy | ||||
| surface tension energy | \(pN \cdot nm\) | output | surfacetension | ||||
| wetting energy | \(pN \cdot nm\) | output | wettingenergy | ||||
| turgor pressure energy | \(pN \cdot nm\) | output | turgorpressure | ||||
| membrane curvature | \(nm^{-1}\) | output | 2÷Ri |
4.114 \(pN \cdot nm\) = 1 \(k_{B}T\) noauthor_kt_2021
1.1. references
2. equations
2.1. summary
arranged by force contributions:
F = membrane tension + bending energy + droplet-cytosol surface tension - droplet-membrane surface tension - droplet-invagination surface tension + turgor pressure\begin{equation} F=\sigma\left(S_{o}+S_{m}+S_{i}\right)+\frac{K}{2} S_{i}\left(\frac{2}{R_{i}}-2C_{i}\right)^{2}+\gamma_{d} S_{d}-\gamma_{m} S_{m}-\gamma_{i} S_{i}+P V_{i} \end{equation}
rearranged by shapes:
\begin{equation} F= \sigma S_{o} + \left(\sigma - \gamma_{m} \right) S_{m} + \gamma_{d} S_{d} + \frac{K}{2} S_{i} \left(\frac{2}{R_{i}} - 2 C_{i}\right)^{2} + P V_{i} + \left(\sigma - \gamma_{i} \right) S_{i} \end{equation}broken down to shape functions and constant \(C\):
\begin{equation} F(\Delta S_{o}, \Delta S_{m}, S_{d}, R_{i}, V_{i})= F_{1}(\Delta S_{o}) + F_{2}(\Delta S_{m}) + F_{3}(S_{d}) + F_{4}(R_{i}) + F_{5}(V_{i}) + C \end{equation} \begin{equation} F_{1}(\Delta S_{o}) = \sigma \Delta S_{o} \end{equation} \begin{equation} \Delta S_{o}=\alpha \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation} \begin{equation} \label{alpha} \alpha = - \left(\frac{\pi\left(\frac{\gamma_{m}}{\gamma_{d}}+1\right)^{3}} {\left(-\frac{\gamma_{m}}{\gamma_{d}}+1\right)\left(\frac{\gamma_{m}}{\gamma_{d}}+2\right)^{2}}\right)^{1 / 3} \end{equation} \begin{equation} \label{} F_{2}(\Delta S_{m}) = \left(\sigma - \gamma_{m} \right) \Delta S_{m} \end{equation} \begin{equation} \Delta S_{m}= - \alpha \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} + S_{i} h^{2} \end{equation} \begin{equation} \label{} F_{3}(S_{d}) = \gamma_{d} S_{d} \end{equation} \begin{equation} S_{d}=\epsilon \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation} \begin{equation} \epsilon = \left(\frac{8 \pi}{\left(-\frac{\gamma_{m}}{\gamma_{d}}+1\right) \left(\frac{\gamma_{m}}{\gamma_{d}}+2\right)^{2}}\right)^{1 / 3} \end{equation} \begin{equation} \label{} \begin{array}{l} F_{4}(R_{i}) = \frac{K}{2} S_{i} \left(\frac{2}{R_{i}} - 2 C_{i}\right)^{2} \end{array} \end{equation} \begin{equation} R_{i} = \frac{\sqrt{S_i}}{2 \sqrt{\pi } h} \end{equation} \begin{equation} \label{} F_{4}(h) = 2 K \left(C_i \sqrt{S_i}-2 \sqrt{\pi } h\right)^{2} \end{equation} \begin{equation} \label{} F_{5}(V_{i}) = P V_{i} \end{equation} \begin{equation} \label{} V_{i}=\eta \left(3 h-2 h^{3}\right) \end{equation} \begin{equation} \label{} \eta=\frac{S_i^{3/2}}{6 \sqrt{\pi }} \end{equation} \begin{equation} \label{} C = \sigma \pi A_t^2+S_i \left(\gamma _m-\gamma _i\right) \end{equation}all combined into full free energy equation:
\begin{equation} F(h)=\left(\gamma_{m} \alpha+\gamma_{d} \epsilon\right)\left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3}+\left(\sigma-\gamma_{m}\right) S_{i} h^{2} +2 K\left(C_{i} \sqrt{S}_{i}-2 \sqrt{\pi} h\right)^{2}+P \eta \left(3 h-2 h^{3}\right)+C \end{equation}expanded polynomial:
\begin{equation} F(h)= \left(\gamma_{m} \alpha+\gamma_{d} \epsilon\right) \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} - 2 P \eta h^{3}+\left(\left(\sigma-\gamma_{m}\right) S_{i}+8 K \pi\right) h^{2} +\left(3 P \eta-8 K C_{i} \sqrt{\pi S_{i}}\right) h+B \end{equation} \begin{equation} B=C+2 K C_{i}^{2} S_{i} \end{equation}2.2. individual shape solutions
2.2.1. relative invagination height
2.2.2. droplet-membrane contact angle
2.2.3. outside membrane area
\begin{equation} \label{outside-membrane-area}
S_{0}=\pi\left(A_{t}^{2}-A_{d}^{2}\right)
\end{equation}
2.2.4. circular plane radii
\begin{equation}
\begin{array}{l}
\sin \theta=\frac{a}{r} \\
a=r \sin \theta} \\
A_{i}=R_{i} \sin \theta_{i}
\end{array}
\end{equation}
\begin{equation} \label{droplet-plane-radius}
A_{d}=R_{d} \sin \theta_{d}
\end{equation}
2.2.5. droplet-cytosol contact area
\begin{equation} \label{dc-contact-area}
\begin{array}{l}
A=2 \pi r^{2}(1-\cos \theta) \\
S_{d}=2 \pi R_{d}^{2}\left(1-\cos \theta_{d}\right)
\end{array}
\end{equation}
2.2.6. invagination surface area
\begin{equation} \label{invagination-area}
S_{i}=4 \pi R_{v}^{2}
\end{equation}
2.2.7. invagination volume
\begin{equation} \label{invagination-volume-a-h}
\begin{array}{l}
V=\frac{1}{6} \pi h\left(3 a^{2}+h^{2}\right) \\
V_{i}=\frac{1}{6} \pi H_{i}\left(3 A_{i}^{2}+H_{i}^{2}\right)
\end{array}
\end{equation}
alternately:
\begin{equation} \label{invagination-volume} \begin{array}{l} V=\frac{\pi}{3} h^{2} \left(3 r - h \right) \\ V_{i}=\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right) \end{array} \end{equation}2.2.8. invagination plane radius
\begin{equation}
\begin{array}{l}
A=\pi\left(a^{2}+h^{2}\right) \\
S_{i}=\pi\left(A_{i}^{2}+H_{i}^{2}\right) \\
A_{i}=\sqrt{\frac{S_{i}}{\pi}-H_{i}^{2}}
\end{array}
\end{equation}
\begin{equation} \label{invagination-plane-radius}
A_{i}=\sqrt{\frac{S_{i}}{\pi} \left(1 - h^{2}\right)}
\end{equation}
2.2.9. invagination radius
\begin{equation} \label{invagination-radius}
\begin{aligned}
A &=2 \pi r h \\
S_{i} &=2 \pi R_{i} H_{i} \\
R_{i} &=\frac{S_{i}}{2 \pi H_{i}}
\end{aligned}
\end{equation}
2.2.10. invagination contact angle
\begin{equation} \label{invagination-angle}
\begin{array}{l}
S_{i}=2 \pi R_{i}^{2}\left(1-\cos \theta_{i}\right) \\
\theta_{i}=\arccos \left(1-\frac{S_{i}}{2 \pi R_{i}^{2}}\right)
\end{array}
\end{equation}
2.2.11. TODO droplet volume
should depend on initial plane radius
\begin{equation} \label{d-volume} V_{b}=V_{d}+V_{i} \end{equation}
2.2.12. droplet + invagination volume
\begin{equation} \label{di-volume}
V_{b}=V_{d}+V_{i}
\end{equation}
2.2.13. droplet + invagination spherical cap radius
\begin{equation} \label{di-cap-radius}
\begin{array}{l}
V=\frac{\pi}{3} r^{3}(2+\cos \theta)(1-\cos \theta)^{2} \\
V_{b}=\frac{\pi}{3} R_{d}^{3}\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2} \\
R_{d}=\left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3}
\end{array}
\end{equation}
2.2.14. droplet + invagination height
\begin{equation} \label{di-height}
\begin{array}{l}
A=2\pi r h \\
S_{d}=2\pi R_{d} H_{d} \\
H_{d}=\frac{S_{d}}{2 \pi R_{d}}
\end{array}
\end{equation}
2.2.15. flat membrane-droplet contact area
\begin{equation} \label{md-area}
S_{m}=\pi\left(A_{d}^{2}-A_{i}^{2}\right)
\end{equation}
2.3. separate force components (original version)
\begin{equation} \label{f-forces}
F=\sigma\left(S_{o}+S_{m}+S_{i}\right)+\frac{K}{2} S_{i}\left(\frac{2}{R_{i}}-2C_{i}\right)^{2}+\gamma_{d} S_{d}-\gamma_{m} S_{m}-\gamma_{i} S_{i}+P V_{i}
\end{equation}
2.4. fully substituted
\begin{equation} \label{free-energy}
F=\sigma(\pi(A_{t}^{2}-(((\frac{3 (V_{d}+(\frac{1}{6} \pi \left(\frac{S_{i} h}{\pi}\right)(3 (\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}+\left(\frac{S_{i} h}{\pi}\right)^{2})))}{\pi(2+(\frac{\gamma_{m}}{\gamma_{d}}))(1-(\frac{\gamma_{m}}{\gamma_{d}}))^{2}})^{1 / 3}) (\sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}))^{2})+(\pi((((\frac{3 (V_{d}+(\frac{1}{6} \pi \left(\frac{S_{i} h}{\pi}\right)(3 (\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}+\left(\frac{S_{i} h}{\pi}\right)^{2})))}{\pi(2+(\frac{\gamma_{m}}{\gamma_{d}}))(1-(\frac{\gamma_{m}}{\gamma_{d}}))^{2}})^{1 / 3}) (\sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}))^{2}-(\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}))+(4 \pi R_{v}^{2}))+
\frac{K}{2} (4 \pi R_{v}^{2})(\frac{2}{(\frac{(4 \pi R_{v}^{2})}{2 \pi \left(\frac{S_{i} h}{\pi}\right)})}-2C_{i})^{2}+
\gamma_{d} (2 \pi R_{d}^{2}(1-(\frac{\gamma_{m}}{\gamma_{d}})))-
\gamma_{m} (\pi((((\frac{3 (V_{d}+(\frac{1}{6} \pi \left(\frac{S_{i} h}{\pi}\right)(3 (\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}+\left(\frac{S_{i} h}{\pi}\right)^{2})))}{\pi(2+(\frac{\gamma_{m}}{\gamma_{d}}))(1-(\frac{\gamma_{m}}{\gamma_{d}}))^{2}})^{1 / 3}) (\sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}))^{2}-(\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}))-
\gamma_{i} (4 \pi R_{v}^{2})+
P (\frac{1}{6} \pi \left(\frac{S_{i} h}{\pi}\right)(3 (\sqrt{\frac{(4 \pi R_{v}^{2})}{\pi}-\left(\frac{S_{i} h}{\pi}\right)^{2}})^{2}+\left(\frac{S_{i} h}{\pi}\right)^{2}))
\end{equation}
2.5. separate shape force components
\begin{equation}
F(\Delta S_{o}, \Delta S_{m}, S_{d}, R_{i}, V_{i})= F_{1}(\Delta S_{o}) + F_{2}(\Delta S_{m}) + F_{3}(S_{d}) + F_{4}(R_{i}) + F_{5}(V_{i}) + C
\end{equation}
\begin{equation} \label{f-shapes}
F= \sigma S_{o} + \left(\sigma - \gamma_{m} \right) S_{m} + \gamma_{d} S_{d} + \frac{K}{2} S_{i} \left(\frac{2}{R_{i}} - 2 C_{i}\right)^{2} + P V_{i} + D
\end{equation}
2.5.1. full derivations
note: there may be some residual messyness since I caught a mistake in the sign of \(\gamma_{d}\) in droplet-contact-angle and corrected it after writing all these equations
- \(F_{1}(\Delta S_{o})\)
\begin{equation} F_{6}(S_{o}) = \sigma S_{o} \end{equation}substituting outside-membrane-area:
\begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-A_{d}^{2}\right) \end{equation}substituting droplet-plane-radius:
\begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-(R_{d} \sin \theta_{d})^{2}\right) \end{equation}substituting di-cap-radius:
\begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-\left(\left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \sin \theta_{d}\right)^{2}\right) \end{equation}substituting droplet-contact-angle:
\begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-\left(\left(\frac{3 V_{b}}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{1 / 3} \sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}\right)^{2}\right) \end{equation}substituting di-volume:
\begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-\left(\left(\frac{3 (V_{d}+V_{i})}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{1 / 3} \sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}\right)^{2}\right) \end{equation}substituting invagination-volume:
\begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-\left(\left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right)\right)}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{1 / 3} \sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}\right)^{2}\right) \end{equation}substituting invagination-radius:
\begin{equation} F_{6}(S_{o}) = \sigma \pi\left(A_{t}^{2}-\left(\left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 \frac{S_{i}}{2 \pi H_{i}} - H_{i} \right)\right)}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{1 / 3} \sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}\right)^{2}\right) \end{equation}"simplified" in mathematica:
\begin{equation} F_{6}(S_{o}) = \sigma \pi \left(A_t^2+\frac{\left(\gamma _m^2-\gamma _d^2\right) \left(\frac{\gamma _d^2 \left(3 V_d+\pi H_i^2\left(\frac{3 S_i}{2 \pi H_i}-H_i\right)\right)}{\pi \left(\frac{\gamma _m}{\gamma _d}+2\right) \left(\gamma _d-\gamma _m\right)^{2}}\right)^{2/3}}{\gamma _d^2}\right) \end{equation}rearranged to isolate constants vs. functions of \(H_{i}\):
replacing invagination height from rel-h:
\begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation} \begin{equation} \Delta S_{o}=\left(\frac{\pi\left(\gamma_{m}^{2}-\gamma_{d}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3}\left(-\pi \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)^{3}+\frac{3}{2} S_{i} h \left(\frac{S_{i}}{\pi}\right)^{1/2}+3 V_{d}\right)^{2 / 3} \end{equation}simplified:
\begin{equation} \Delta S_{o}=\left(\frac{\pi\left(\gamma_{m}^{2}-\gamma_{d}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3}\left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation} \begin{equation} \Delta S_{o}=\alpha \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} \end{equation} \begin{equation} \alpha = \left(\frac{\pi\left(\gamma_{m}^{2}-\gamma_{d}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3} \end{equation} - \(F_{2}(\Delta S_{m})\)
gathering from free-energy:
\begin{equation} \label{} F_{7}(S_{m}) = \left(\sigma - \gamma_{m} \right) S_{m} \end{equation}from md-area:
\begin{equation} \label{md-area} S_{m}=\pi\left(A_{d}^{2}-A_{i}^{2}\right) \end{equation}substitute droplet-plane-radius:
\begin{equation} A_{d}=R_{d} \sin \theta_{d} \end{equation}and invagination-plane-radius:
\begin{equation} A_{i}=\sqrt{\frac{S_{i}}{\pi}-H_{i}^{2}} \end{equation}to get:
\begin{equation} S_{m}=\pi\left(\left(R_{d} \sin \theta_{d}\right)^{2}-\frac{S_{i}}{\pi}+H_{i}^{2}\right) \end{equation}rearranged:
\begin{equation} S_{m}=\pi\left(\left(R_{d} \sin \theta_{d}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}substitute di-cap-radius:
\begin{equation} R_{d}=\left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \end{equation}to get:
\begin{equation} S_{m}=\pi\left(\left(\left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \sin \theta_{d}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}substitute di-volume:
\begin{equation} V_{b}=V_{d}+V_{i} \end{equation}to get:
\begin{equation} S_{m}=\pi\left(\left(\left(\frac{3 \left(V_{d}+V_{i} \right)}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \sin \theta_{d}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}substitute invagination-volume:
\begin{equation} V_{i}=\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right) \end{equation}to get:
\begin{equation} S_{m}=\pi\left(\left(\left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right) \right)}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \sin \theta_{d}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}substitute invagination-radius:
\begin{equation} R_{i} =\frac{S_{i}}{2 \pi H_{i}} \end{equation}to get:
\begin{equation} S_{m}=\pi\left(\left(\left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 \frac{S_{i}}{2 \pi H_{i}} - H_{i} \right) \right)}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \sin \theta_{d}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}substitute droplet-contact-angle:
\begin{equation} \begin{array}{l} \cos \theta_{d}=\frac{\gamma_{m}}{\gamma_{d}} \\ \sin \theta_{d}=\sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}} \end{array} \end{equation}to get:
\begin{equation} S_{m}=\pi\left(\left(\left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 \frac{S_{i}}{2 \pi H_{i}} - H_{i} \right) \right)}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{1 / 3} \sqrt{1 - (\frac{\gamma_{m}}{\gamma_{d}})^{2}}\right)^{2}+H_{i}^{2}\right) - S_{i} \end{equation}simplified in mathematica:
\begin{equation} S_{m}=\pi \left(\frac{\left(\gamma _d^2-\gamma _m^2\right) \left(\frac{\gamma _d^2 \left(3 V_d+\pi H_i^2\left(\frac{3 S_i}{2 \pi H_i}-H_i\right)\right)}{\pi \left(2-\frac{\gamma _m}{\gamma _d}\right) \left(\gamma _d+\gamma _m\right)^{2}}\right)^{2/3}}{\gamma _d^2}+H_i^2\right)-S_i \end{equation}manually rearranged:
replacing invagination height from rel-h:
\begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation} \begin{equation} \Delta S_{m}=\left(\frac{\pi\left(\gamma_{d}^{2}-\gamma_{m}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3}\left(-\pi \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)^{3}+\frac{3}{2} S_{i} \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)+3 V_{d}\right)^{2 / 3}+ \pi \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)^{2} \end{equation}simplified:
\begin{equation} \Delta S_{m}=\left(\frac{\pi\left(\gamma_{d}^{2}-\gamma_{m}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3} \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} + S_{i} h^{2} \end{equation} \begin{equation} \Delta S_{m}= \beta \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} + S_{i} h^{2} \end{equation} \begin{equation} \beta = \left(\frac{\pi\left(\gamma_{d}^{2}-\gamma_{m}^{2}\right)^{3}}{\left(2 \gamma_{d}+\gamma_{m}\right)^{2}\left(\gamma_{d}-\gamma_{m}\right)^{4}}\right)^{1 / 3} \end{equation} - \(F_{3}(S_{d})\)
gathering from free-energy:
\begin{equation} \label{} F_{3}(S_{d}) = \gamma_{d} S_{d} \end{equation}from dc-contact-area:
\begin{equation} S_{d}=2 \pi R_{d}^{2}\left(1-\cos \theta_{d}\right) \end{equation}substitute di-cap-radius
\begin{equation} R_{d}=\left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{1 / 3} \end{equation}to get:
\begin{equation} S_{d}=2 \pi \left(\frac{3 V_{b}}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{2 / 3}\left(1-\cos \theta_{d}\right) \end{equation}substitute di-volume
\begin{equation} V_{b}=V_{d}+V_{i} \end{equation}to get:
\begin{equation} S_{d}=2 \pi \left(\frac{3 \left(V_{d}+V_{i}\right)}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{2 / 3}\left(1-\cos \theta_{d}\right) \end{equation}substitute invagination-volume
\begin{equation} V_{i}=\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right) \end{equation}to get:
\begin{equation} S_{d}=2 \pi \left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right)\right)}{\pi\left(2+\cos \theta_{d}\right)\left(1-\cos \theta_{d}\right)^{2}}\right)^{2 / 3}\left(1-\cos \theta_{d}\right) \end{equation}substitute droplet-contact-angle
\begin{equation} \cos \theta_{d}=\frac{\gamma_{m}}{\gamma_{d}} \end{equation}to get:
\begin{equation} S_{d}=2 \pi \left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 R_{i} - H_{i} \right)\right)}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{2 / 3}\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right) \end{equation}substitute invagination-radius
\begin{equation} R_{i} &=\frac{S_{i}}{2 \pi H_{i}} \end{equation}to get:
\begin{equation} S_{d}=2 \pi \left(\frac{3 \left(V_{d}+\frac{\pi}{3} H_{i}^{2} \left(3 \frac{S_{i}}{2 \pi H_{i}} - H_{i} \right)\right)}{\pi\left(2+\frac{\gamma_{m}}{\gamma_{d}}\right)\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right)^{2}}\right)^{2 / 3}\left(1-\frac{\gamma_{m}}{\gamma_{d}}\right) \end{equation}simplified in mathematica:
\begin{equation} S_{d}= \frac{2 \pi \left(\gamma _d-\gamma _m\right) \left(\frac{\gamma _d^2 \left(3 V_d+\pi H_i^2\left(\frac{3 S_i}{2 \pi H_i}-H_i\right)\right)}{\pi \left(\frac{\gamma _m}{\gamma _d}+2\right) \left(\gamma _d-\gamma _m\right)^{2}}\right)^{2/3}}{\gamma _d} \end{equation}manually rearranged:
- \(F_{4}(R_{i})\)
gathering from f-forces:
\begin{equation} \label{} \begin{array}{l} F_{4}(R_{i}) = \frac{K}{2} S_{i} \left(\frac{2}{R_{i}} - 2 C_{i}\right)^{2} \end{array} \end{equation}from invagination-radius
\begin{equation} R_{i} &=\frac{S_{i}}{2 \pi H_{i}} \end{equation}replacing invagination height from rel-h:
\begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation} \begin{equation} R_{i} = \frac{S_{i}}{2 \pi h \left(\frac{S_{i}}{\pi}\right)^{1/2}} \end{equation}simplified:
\begin{equation} R_{i} = \frac{\sqrt{S_i}}{2 \sqrt{\pi } h} \end{equation}- \(F_{4}(h)\)
substitute invagination-radius
\begin{equation} R_{i} &=\frac{S_{i}}{2 \pi H_{i}} \end{equation}to get:
\begin{equation} \label{} \begin{array}{l} F_{4}(H_{i}) = \frac{K}{2} S_{i} \left(\frac{2}{\frac{S_{i}}{2 \pi H_{i}}} - 2 C_{i}\right)^{2} \end{array} \end{equation}simplified:
\begin{equation} \label{} F_{4}(H_{i}) = \frac{2 K}{S_i} \left(C_i S_i-2 \pi H_i\right)^{2} \end{equation}replacing invagination height from rel-h:
\begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation} \begin{equation} \label{} F_{4}(h) = \frac{2 K}{S_i} \left(C_i S_i-2 \pi \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)\right)^{2} \end{equation}simplified:
\begin{equation} \label{} F_{4}(h) = 2 K \left(C_i \sqrt{S_i}-2 \sqrt{\pi } h\right)^2 \end{equation}
- \(F_{4}(h)\)
- \(F_{5}(V_{i})\)
gathering from f-forces:
\begin{equation} \label{} F_{5}(V_{i}) = P V_{i} \end{equation}from invagination-volume-a-h:
\begin{equation} \label{} \begin{array}{l} V_{i}=\frac{1}{6} \pi H_{i}\left(3 A_{i}^{2}+H_{i}^{2}\right) \end{array} \end{equation}substitute invagination-plane-radius
\begin{equation} \label{} \begin{array}{l} A_{i}=\sqrt{\frac{S_{i}}{\pi}-H_{i}^{2}} \end{array} \end{equation}to get:
\begin{equation} \label{} \begin{array}{l} V_{i}=\frac{1}{6} \pi H_{i}\left(3 \left(\frac{S_{i}}{\pi}-H_{i}^{2}\right)+H_{i}^{2}\right) \end{array} \end{equation}simplified:
\begin{equation} \label{} \begin{array}{l} V_{i}=\frac{1}{6} \pi H_{i}\left(\frac{3 S_i}{\pi }-2 H_{i}^2\right) \end{array} \end{equation}replacing invagination height from rel-h:
\begin{equation} H_{i} = h \left(\frac{S_{i}}{\pi}\right)^{1/2} \end{equation} \begin{equation} \label{} V_{i}=\frac{1}{6} \pi h \left(\frac{S_{i}}{\pi}\right)^{1/2}\left(\frac{3 S_i}{\pi }-2 \left(h \left(\frac{S_{i}}{\pi}\right)^{1/2}\right)^2\right) \end{equation}simplified:
\begin{equation} \label{} V_{i}= \frac{h \left(3-2 h^2\right) S_i^{3/2}}{6 \sqrt{\pi }} \end{equation} \begin{equation} \label{} V_{i}=\eta \left(3 h-2 h^{3}\right) \end{equation} \begin{equation} \label{} \eta=\frac{S_i^{3/2}}{6 \sqrt{\pi }} \end{equation} - \(D\): everything else that doesn't change with \(H_{i}\)
\begin{equation} \label{D} D = \sigma S_{i} - \gamma_{i} S_{i} \end{equation} - \(C\): additional constants grouped out from \(\Delta\) terms
\begin{equation} \label{} C = D + \sigma \pi A_{t}^{2} - \left(\sigma - \gamma_{m} \right) S_{i} \end{equation} \begin{equation} \label{} D = \sigma S_{i} - \gamma_{i} S_{i} \end{equation}simplified:
\begin{equation} \label{} C = \sigma \pi A_t^2+S_i \left(\gamma _m-\gamma _i\right) \end{equation}
2.5.2. DONE substituted after isolating \(h\)
- could anything be factored out more first?
2.5.3. testing sign of alpha terms
import numpy as np import math def cube_root(i): return math.pow(abs(i),float(1)/3) * (1,-1)[i<0] def alpha_so(gamma_m, gamma_d): alpha = cube_root((np.pi*(gamma_m**2 - gamma_d**2)**3)/((2*gamma_d+gamma_m)**2 * (gamma_d-gamma_m)**4)) return alpha for gamma_m in np.linspace(0, 200, 9): for gamma_d in np.linspace(0, 200, 9): print('gamma_m',gamma_m,'gamma_d',gamma_d,alpha_so(gamma_m, gamma_d))
gamma_m 0.0 gamma_d 0.0 nan gamma_m 0.0 gamma_d 25.0 -0.9226350743220142 gamma_m 0.0 gamma_d 50.0 -0.9226350743220142 gamma_m 0.0 gamma_d 75.0 -0.9226350743220142 gamma_m 0.0 gamma_d 100.0 -0.9226350743220142 gamma_m 0.0 gamma_d 125.0 -0.9226350743220142 gamma_m 0.0 gamma_d 150.0 -0.9226350743220142 gamma_m 0.0 gamma_d 175.0 -0.9226350743220142 gamma_m 0.0 gamma_d 200.0 -0.9226350743220142 gamma_m 25.0 gamma_d 0.0 1.4645918875615231 gamma_m 25.0 gamma_d 25.0 nan gamma_m 25.0 gamma_d 50.0 -1.5026501396568157 gamma_m 25.0 gamma_d 75.0 -1.2706753068765375 gamma_m 25.0 gamma_d 100.0 -1.1735039002840684 gamma_m 25.0 gamma_d 125.0 -1.1192302015380546 gamma_m 25.0 gamma_d 150.0 -1.084415611760493 gamma_m 25.0 gamma_d 175.0 -1.0601370721917294 gamma_m 25.0 gamma_d 200.0 -1.042222647325497 gamma_m 50.0 gamma_d 0.0 1.4645918875615231 gamma_m 50.0 gamma_d 25.0 1.7436710272644398 gamma_m 50.0 gamma_d 50.0 nan gamma_m 50.0 gamma_d 75.0 -1.830739859451904 gamma_m 50.0 gamma_d 100.0 -1.5026501396568157 gamma_m 50.0 gamma_d 125.0 -1.356188576761231 gamma_m 50.0 gamma_d 150.0 -1.2706753068765375 gamma_m 50.0 gamma_d 175.0 -1.214010595446712 gamma_m 50.0 gamma_d 200.0 -1.1735039002840684 gamma_m 75.0 gamma_d 0.0 1.4645918875615231 gamma_m 75.0 gamma_d 25.0 1.590205608287594 gamma_m 75.0 gamma_d 50.0 2.001188208394222 gamma_m 75.0 gamma_d 75.0 nan gamma_m 75.0 gamma_d 100.0 -2.0727783992021025 gamma_m 75.0 gamma_d 125.0 -1.6820324801561286 gamma_m 75.0 gamma_d 150.0 -1.5026501396568157 gamma_m 75.0 gamma_d 175.0 -1.3955026949998752 gamma_m 75.0 gamma_d 200.0 -1.3231738439535343 gamma_m 100.0 gamma_d 0.0 1.4645918875615231 gamma_m 100.0 gamma_d 25.0 1.5377251238700236 gamma_m 100.0 gamma_d 50.0 1.7436710272644398 gamma_m 100.0 gamma_d 75.0 2.208757298511275 gamma_m 100.0 gamma_d 100.0 nan gamma_m 100.0 gamma_d 125.0 -2.2692052337015594 gamma_m 100.0 gamma_d 150.0 -1.830739859451904 gamma_m 100.0 gamma_d 175.0 -1.6263744927117951 gamma_m 100.0 gamma_d 200.0 -1.5026501396568157 gamma_m 125.0 gamma_d 0.0 1.4645918875615231 gamma_m 125.0 gamma_d 25.0 1.512803489134373 gamma_m 125.0 gamma_d 50.0 1.6429054603976958 gamma_m 125.0 gamma_d 75.0 1.8801889207945017 gamma_m 125.0 gamma_d 100.0 2.384131644400035 gamma_m 125.0 gamma_d 125.0 nan gamma_m 125.0 gamma_d 150.0 -2.436744690673985 gamma_m 125.0 gamma_d 175.0 -1.959079850097313 gamma_m 125.0 gamma_d 200.0 -1.7343631139416589 gamma_m 150.0 gamma_d 0.0 1.4645918875615231 gamma_m 150.0 gamma_d 25.0 1.498872430465395 gamma_m 150.0 gamma_d 50.0 1.590205608287594 gamma_m 150.0 gamma_d 75.0 1.7436710272644396 gamma_m 150.0 gamma_d 100.0 2.001188208394222 gamma_m 150.0 gamma_d 125.0 2.5372464543855386 gamma_m 150.0 gamma_d 150.0 nan gamma_m 150.0 gamma_d 175.0 -2.5840841134673402 gamma_m 150.0 gamma_d 200.0 -2.0727783992021025 gamma_m 175.0 gamma_d 0.0 1.4645918875615234 gamma_m 175.0 gamma_d 25.0 1.4902570606397723 gamma_m 175.0 gamma_d 50.0 1.5585019217103644 gamma_m 175.0 gamma_d 75.0 1.6687875802778238 gamma_m 175.0 gamma_d 100.0 1.836572392913886 gamma_m 175.0 gamma_d 125.0 2.1098678647384412 gamma_m 175.0 gamma_d 150.0 2.6739764366927052 gamma_m 175.0 gamma_d 175.0 nan gamma_m 175.0 gamma_d 200.0 -2.71637250521393 gamma_m 200.0 gamma_d 0.0 1.4645918875615231 gamma_m 200.0 gamma_d 25.0 1.4845441581729593 gamma_m 200.0 gamma_d 50.0 1.5377251238700236 gamma_m 200.0 gamma_d 75.0 1.6219368867750477 gamma_m 200.0 gamma_d 100.0 1.7436710272644398 gamma_m 200.0 gamma_d 125.0 1.9220789459321215 gamma_m 200.0 gamma_d 150.0 2.208757298511275 gamma_m 200.0 gamma_d 175.0 2.7980755038451366 gamma_m 200.0 gamma_d 200.0 nan /tmp/ipykernel_172232/638675132.py:7: RuntimeWarning: invalid value encountered in double_scalars alpha = cube_root((np.pi*(gamma_m**2 - gamma_d**2)**3)/((2*gamma_d+gamma_m)**2 * (gamma_d-gamma_m)**4)) /tmp/ipykernel_172232/638675132.py:4: DeprecationWarning: In future, it will be an error for 'np.bool_' scalars to be interpreted as an index return math.pow(abs(i),float(1)/3) * (1,-1)[i<0]
3. implementation and analysis [36/42]
3.1. analytical derivations
3.1.1. thermodynamic spontaneity (global stability) [5/5]
- I'm defining "thermodynamic spontaneity" as whether F(h=0) > F(h=1)
- analyzed by plotting the phase boundary line across 2 parameters when F(h=0) == F(h=1)
- DONE setting up equality
- DONE solving for \(K\)
- DONE solving for \(P\)
- DONE solving for \(\sigma\)
- DONE solving for \(C_{i}\)
- any others?
3.1.2. kinetic spontaneity (local stability) [11/11]
- I'm defining "kinetic spontaneity" as whether F'(h=0) is negative or positive
- DONE solving derivative
- DONE solving for h=0
\begin{equation} \label{f-prime-alpha-beta} F^{\prime}(0)=\frac{\left(\sigma \alpha+\left(\sigma-\gamma_{m}\right) \beta+\gamma_{d} \epsilon\right) S_{i}^{3 / 2}}{\sqrt{\pi} \left(3 V_{d}\right)^{1 / 3}}+3 P \eta-8 K C_{i} \sqrt{\pi S_{i}} \end{equation} \begin{equation} \label{f-prime} F^{\prime}(0)=\frac{\left(\gamma_{m} \alpha + \gamma_{d} \epsilon \right) S_{i}^{3 / 2}}{\sqrt{\pi} \left(3 V_{d}\right)^{1 / 3}}+3 P \eta-8 K C_{i} \sqrt{\pi S_{i}} \end{equation} - DONE sign of droplet prefactor at h=0
the sign of the term concerning the droplet must always be positive.
- DONE solving for \(K\)
- DONE solving for \(P\)
- DONE solving for \(\sigma\)
- DONE solving for \(C_{i}\)
- CANCELLED solving for \(\gamma_{m}\)
- CANCELLED solving for \(\gamma_{d}\)
- CANCELLED solving for \(S_{i}\)
- DONE solving for \(V_{d}\)
3.2. python code [20/26]
3.2.1. configuration [12/13]
- import libraries, settings
%gui qt import napari from decimal import Decimal import math import numpy as np import pandas as pd import matplotlib.pyplot as plt # plotting import mpl_toolkits.mplot3d.axes3d as axes3d %matplotlib inline import seaborn as sns import scipy #sns.set(style='ticks') #sns.set_palette(sns.cubehelix_palette(10, start=.1, rot=-.75, reverse=True)) sns.set_palette('hls',8) #sns.set_palette(sns.diverging_palette(250, 30, l=65, center="dark")) # sns.set_palette("coolwarm", 6) # from cycler import cycler SMALL_SIZE = 12 MEDIUM_SIZE = 16 BIGGER_SIZE = 20 plt.rc('axes', titlesize=MEDIUM_SIZE) # fontsize of the axes title plt.rc('axes', labelsize=MEDIUM_SIZE) # fontsize of the x and y labels plt.rc('xtick', labelsize=SMALL_SIZE) # fontsize of the tick labels plt.rc('ytick', labelsize=SMALL_SIZE) # fontsize of the tick labels plt.rc('legend', fontsize=SMALL_SIZE) # legend fontsize plt.rc('figure', titlesize=BIGGER_SIZE) # fontsize of the figure title # default_cycler = (cycler(color=['indigo', 'darkgreen', 'purple', 'green', 'magenta', 'lime'])) # plt.rc('axes', prop_cycle=default_cycler) pi = np.pi #if kernel was started in notebook directory #fig_dir = 'figures/' #assuming kernel was started in home directory fig_dir = 'google_drive/grad_school/db_lab/code/analysis/llps_endocytosis_modeling/figures/'
:results: nil:END:
import IPython from tabulate import tabulate class OrgFormatter(IPython.core.formatters.BaseFormatter): def __call__(self, obj): try: return tabulate(obj, headers='keys', tablefmt='orgtbl', showindex='always') except: return None ip = get_ipython() ip.display_formatter.formatters['text/org'] = OrgFormatter()
:results: nil:END:
- set default fixed parameter values
- assigned here
vesicle_radius = 50. #nm vesicle_diameter = 2*vesicle_radius #nm drop_radius = 1500. #nm S_i = 4.*pi*vesicle_radius**2 #nm^2 #H_i_range = np.linspace(0.001*vesicle_radius,2*vesicle_radius,1000) #nm h_range = np.linspace(0.001,1,1000) #dimensionless 0-1 H_i_range = h_range*(S_i/pi)**(1/2) sigma = 150. #arbitrary, N/nm^2 A_t = 1000 #nm, arbitrary K = 100. #arbitrary, N/nm C_i = 1/vesicle_radius #nm^-1 gamma_d = 200. #arbitrary N/nm^2 gamma_m = 175. #arbitrary, N/nm^2 gamma_i = gamma_m V_d = 4.*pi*drop_radius**2 #nm^3 P = 1000/(10**9) #N/nm^3 params = {'vesicle_radius' : vesicle_radius, 'vesicle_diameter' : vesicle_diameter, 'drop_radius' : drop_radius, 'S_i' : S_i, 'h_range' : h_range, 'H_i_range' : H_i_range, 'sigma' : sigma, 'A_t' : A_t, 'K' : K, 'C_i' : C_i, 'gamma_d' : gamma_d, 'gamma_m' : gamma_m, 'gamma_i' : gamma_i, 'V_d' : V_d, 'P':P}
:results: nil:END:
- assigned from org table
established in variables
variable_df = pd.DataFrame(variable_list[1:], columns=variable_list[0]) variable_df = variable_df.set_index('python variable') param_values = variable_df['value'] variable_df
python variable description value units reference type associated equation LaTeX variable Hi invagination height \(nm\) input (variable) \(H_{i}\) sigma membrane tension 0.001 \(pN/nm\) hassinger_design_2017 fixed \(\sigma\) At total plane radius 2000 \(nm\) fixed \(A_{t}\) vesicleradius final vesicle radius 50 \(nm\) kaksonen_mechanisms_2018 fixed \(R_{v}\) K coat bending rigidity 160 \(pN nm\) dmitrieff_membrane_2015 fixed \(K\) Ci spontaneous curvature 1/50 \(nm^{-1}\) hassinger_design_2017 fixed \(C_{i}\) gammad droplet-cytosol surface tension 0.005 \(pN/nm\) jawerth_salt-dependent_2018 fixed \(\gamma_{d}\) gammam droplet-membrane wetting tension 0.002 \(pN/nm\) fixed \(\gamma_{m}\) initial droplet plane radius fixed gammai droplet-invagination surface tension 0.002 \(pN/nm\) fixed \(\gamma_{i}\) P turgor pressure 0.1 \(pN/nm^2\) ma_mechanics_2019 fixed \(P\) h relative invagination height unitless input (variable) rel-h \(h\) Vd droplet volume \(nm^3\) intermediate output \(V_{d}\) Si invagination surface area \(nm^2\) intermediate output invagination-area \(S_{i}\) dropradius initial droplet spherical radius 600 \(nm\) intermediate output Sm droplet-flat membrane contact area \(nm^2\) intermediate output md-area \(S_{m}\) Ri invagination spherical radius \(nm\) intermediate output invagination-radius \(R_{i}\) Sd droplet-cytosol contact area \(nm^2\) intermediate output dc-contact-area \(S_{d}\) thetad droplet contact angle rad intermediate output droplet-contact-angle \(\theta_{d}\) thetai invagination contact angle rad intermediate output invagination-angle \(\theta_{i}\) Ai invagination plane radius \(nm\) intermediate output invagination-plane-radius \(A_{i}\) Ad droplet + invagination plane radius \(nm\) intermediate output droplet-plane-radius \(A_{d}\) So outside membrane surface area \(nm^2\) intermediate output outside-membrane-area \(S_{o}\) Vi invagination volume \(nm^3\) intermediate output invagination-volume \(V_{i}\) Vb droplet + invagination volume \(nm^3\) intermediate output di-volume \(V_{b}\) Hd droplet + invagination height \(nm\) intermediate output di-height \(H_{d}\) Rd droplet + invagination spherical radius \(nm\) intermediate output di-cap-radius \(R_{d}\) F free energy \(pN \cdot nm\) output f-forces f-shapes \(F\) membranetension membrane tension energy \(pN \cdot nm\) output bendingenergy bending energy \(pN \cdot nm\) output surfacetension surface tension energy \(pN \cdot nm\) output wettingenergy wetting energy \(pN \cdot nm\) output turgorpressure turgor pressure energy \(pN \cdot nm\) output 2÷Ri membrane curvature \(nm^{-1}\) output vesicle_radius = param_values['vesicle_radius'] vesicle_diameter = 2*vesicle_radius #nm drop_radius = param_values['drop_radius'] #nm S_i = 4.*pi*vesicle_radius**2 #nm^2 #H_i_range = np.linspace(0.001*vesicle_radius,2*vesicle_radius,1000) #nm h_range = np.linspace(0.001,1,1000) #dimensionless 0-1 H_i_range = h_range*(S_i/pi)**(1/2) sigma = param_values['sigma'] #arbitrary, N/nm^2 A_t = param_values['A_t'] #nm, arbitrary K = param_values['K'] #arbitrary, N/nm C_i = 1/vesicle_radius #nm^-1 gamma_d = param_values['gamma_d'] #arbitrary N/nm^2 gamma_m = param_values['gamma_m'] #arbitrary, N/nm^2 gamma_i = gamma_m V_d = 4.*pi*drop_radius**2 #nm^3 P = param_values['P'] #N/nm^2 params = {'vesicle_radius' : vesicle_radius, 'vesicle_diameter' : vesicle_diameter, 'drop_radius' : drop_radius, 'S_i' : S_i, 'h_range' : h_range, 'H_i_range' : H_i_range, 'sigma' : sigma, 'A_t' : A_t, 'K' : K, 'C_i' : C_i, 'gamma_d' : gamma_d, 'gamma_m' : gamma_m, 'gamma_i' : gamma_i, 'V_d' : V_d, 'P':P}
:results: nil:END:
- assigned here
- define composite variables
This is just to check what they are, I re-define them whenever I solve for F
alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) gamma_ratio = gamma_m/gamma_d alpha_refactored = np.cbrt((pi*(gamma_ratio+1)**3)/((gamma_ratio-1)*(gamma_ratio+2)**2)) beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) #epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) epsilon = np.cbrt((8*pi)/((2+gamma_ratio)**2*(1-gamma_ratio))) eta = (S_i**(3/2))/(6*pi**(1/2)) C = sigma*pi*A_t**2+S_i*(gamma_m-gamma_i) alpha, alpha_refactored, beta, epsilon, eta, C
-1.3561885767612312 -1.356188576761231 1.3561885767612312 1.937412252516044 523598.7755982989 12566.370614359173 - custom functions
[12/13]
- model functions
- pre-analytical version
[ ]rewrite each function variable as dictionary key (outpus['theta_d']instead oftheta_dfor the first one) and just return outputs at endshould eliminate some redundancy
def llps_model(H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P): #H_i = h*(S_i/pi)**(1/2) # might need to be -gamma_m/gamma_d theta_d = np.arccos(gamma_m/gamma_d) #equation 8 #to make a full array out of the constant theta_d value #theta_d = np.full_like(H_i, theta_d) A_i = ((S_i/pi)-H_i**2)**(1/2) #equation 7 V_i = (1/6)*pi*H_i*(3*A_i**2+H_i**2) #equation 6 # this is actually wrong! # R_i = (A_i**2-H_i**2)/(2*H_i) #equation 10 R_i = S_i/(2*pi*H_i) #equation 10 re-do # this is wrong! # theta_i = np.arcsin(A_i/R_i) #equation 12 theta_i = np.arccos(1-S_i/(2*pi*R_i**2)) #equation 12 re-do V_b = V_d+V_i # equation 5 # R_d_list = [] # for V in V_b: # R_d_list.append(cube_root((3*V)/(pi*(2+np.cos(theta_d))*(1-np.cos(theta_d))**2))) R_d = ((3*V_b)/(pi*(2+np.cos(theta_d))*(1-np.cos(theta_d))**2))**(1/3) #equation 4 # R_d = np.array(R_d_list) S_d = 2*pi*R_d**2*(1-np.cos(theta_d)) #equation 11 A_d = R_d*np.sin(theta_d) #equation 3 S_m = pi*(A_d**2-A_i**2) #equation 9 S_o = pi*(A_t**2-A_d**2) #equation 2 membrane_tension = sigma*(S_o+S_m+S_i) bending_energy = (K/2)*S_i*((2/R_i)-(2*C_i))**2 surface_tension = gamma_d*S_d wetting_energy = gamma_m*S_m+gamma_i*S_i droplet_energy = surface_tension - wetting_energy turgor_pressure = P*V_i F = membrane_tension+bending_energy+surface_tension-wetting_energy+turgor_pressure #equation 1 outputs = {'theta_d':theta_d,'A_i':A_i,'V_i':V_i, 'R_i':R_i, '2÷R_i':2/R_i,'theta_i':theta_i,'V_b':V_b,'R_d':R_d, 'S_d':S_d,'A_d':A_d,'S_m':S_m,'S_o':S_o, 'F':F,'membrane_tension':membrane_tension, 'bending_energy':bending_energy, 'bending_energy_flipped':-bending_energy, 'surface_tension':surface_tension, 'wetting_energy':-wetting_energy, 'wetting_energy_flipped':wetting_energy, 'droplet_energy':droplet_energy, 'turgor_pressure':turgor_pressure} return outputs
nil:END:
plt.plot(H_i_range/100,llps_model()['F']) plt.xlabel('h') plt.ylabel('F') plt.tight_layout() plt.savefig(fig_dir+'llps_model_test.png')
:results:
- post-analytical version
def llps_model_recbrt(h=h_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P): alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) eta = (S_i**(3/2))/(6*pi**(1/2)) C = sigma*pi*A_t**2+S_i*(gamma_m-gamma_i) F = ((sigma*alpha+(sigma-gamma_m)*beta+gamma_d*epsilon)* np.cbrt(((S_i**(3/2))/(pi**(1/2)))*(3*h/2-h**3)+3*V_d)**2+ (sigma-gamma_m)*S_i*h**2+ 2*K*(C_i*S_i**(1/2)-2*pi**(1/2)*h)**2+ P*eta*(3*h-2*h**3)+ C ) # print(alpha, beta, epsilon, eta, C) return F
nil:END:
plt.plot(h_range,llps_model_recbrt()) plt.xlabel('h') plt.ylabel('F') plt.tight_layout() plt.savefig(fig_dir+'llps_model_recbrt_test.png')
:results:
- pre-analytical version
- general plotting function
- figure out some way to get the right units for each thing being plotted
def plot_simulation(x=H_i_range/vesicle_diameter, output='F', H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P, label=None, variable_df=variable_df): outputs = llps_model(H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) y = outputs[output] plt.plot(x,y,label=label) plt.ylabel(variable_df.loc[output]['description']+' ('+variable_df.loc[output]['units']+')')
:results: nil:END:
- normalized plots
def plot_simulation_normalized(x=H_i_range/vesicle_diameter, output='F', H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P, label=None): outputs = llps_model(H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d) y_sub = outputs[output]-np.min(outputs[output]) y_norm = y_sub/np.max(y_sub) plt.plot(x,y_norm,label=label) plt.ylabel(output) plt.xlabel('invagination distance/vesicle diameter')
:results: nil:END:
- delta plots
def plot_simulation_delta(x=H_i_range/vesicle_diameter, output='F', H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P, label=None, variable_df=variable_df, color=None): outputs = llps_model(H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) output_0 = llps_model(H_i=H_i_range[0], sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) y = outputs[output]-output_0[output] plt.plot(x,y,label=label, color=color) #plt.ylabel('∆ '+variable_df.loc[output]['description']+' ('+variable_df.loc[output]['units']+')') plt.xlabel('h: invagination distance/vesicle diameter')
:results: nil:END:
- show geometry for given invagination distance
def show_geometry(H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P, fig_width=10, fig_height=5, num_plots=1, top=20, bottom=-250, left=-500, right=500): outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) A_d = outputs['A_d'] A_i = outputs['A_i'] theta_d = outputs['theta_d'] theta_i = outputs['theta_i'] R_d = outputs['R_d'] R_i = outputs['R_i'] #fig, ax = plt.subplots(1,num_plots) #plt.subplot(1, num_plots, 1) fig = plt.figure() fig.set_size_inches((fig_width,fig_height)) ax = fig.add_subplot(1, num_plots, 1) plt.hlines([0,0],[-A_t,A_d],[-A_d,A_t], color='r', label='$S_o$') plt.hlines([0,0],[-A_d,A_i],[-A_i,A_d], color='g', label='$S_m$') if V_d > 0: points_d = np.linspace(pi/2-theta_d, theta_d+pi/2, 1000) x_d = R_d*np.cos(points_d) y_d = R_d*np.sin(points_d) y_d_adj = y_d-np.min(y_d) plt.plot(x_d,-y_d_adj, color='b', label='$S_d$') points_i = np.linspace(pi/2-theta_i, theta_i+pi/2, 1000) x_i = R_i*np.cos(points_i) y_i = R_i*np.sin(points_i) y_i_adj = y_i-np.min(y_i) plt.plot(x_i,-y_i_adj, color='purple', label='$S_i$') ax.set_ylim(top=top, bottom=bottom) ax.set_xlim(left=left, right=right) ax.set_aspect('equal') ax.set(xlabel = 'distance along membrane (nm)', ylabel = 'distance into cytoplasm (nm)') plt.legend() plt.tight_layout() if num_plots > 1: return fig
:results: nil:END:
- 3D geometry
- inspired by noauthor_python_nodate
- need to turn this into a function
- points
%gui qt viewer = napari.Viewer(ndisplay=3) spherecoords = [] for H_i in np.linspace(1,100,100): outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) A_d = outputs['A_d'] A_i = outputs['A_i'] theta_d = outputs['theta_d'] theta_i = outputs['theta_i'] R_d = outputs['R_d'] R_i = outputs['R_i'] theta, phi = np.linspace(0,2* np.pi, 200), np.linspace(pi-theta_d, np.pi, 200) THETA, PHI = np.meshgrid(theta, phi) R = R_d X = R * np.sin(PHI) * np.cos(THETA) Y = R * np.sin(PHI) * np.sin(THETA) Z = R * np.cos(PHI) Z = Z-np.max(Z) height = np.full_like(Z, H_i) spherecoords.append(np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T) spherecoords = np.concatenate(spherecoords) viewer.add_points(np.array(spherecoords), name='droplet', size=20, face_color='cyan', edge_color = 'transparent', blending='additive', opacity=0.6) spherecoords = [] for H_i in np.linspace(1,100,100): outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) A_d = outputs['A_d'] A_i = outputs['A_i'] theta_d = outputs['theta_d'] theta_i = outputs['theta_i'] R_d = outputs['R_d'] R_i = outputs['R_i'] theta, phi = np.linspace(0,2* np.pi, 200), np.linspace(pi-theta_i, np.pi, 200) THETA, PHI = np.meshgrid(theta, phi) R = R_i X = R * np.sin(PHI) * np.cos(THETA) Y = R * np.sin(PHI) * np.sin(THETA) Z = R * np.cos(PHI) Z = Z-np.max(Z) height = np.full_like(Z, H_i) spherecoords.append(np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T) spherecoords = np.concatenate(spherecoords) viewer.add_points(np.array(spherecoords), name='coat', size=20, face_color='purple', edge_color = 'transparent', blending='additive', opacity=0.6) spherecoords = [] for H_i in np.linspace(1,100,100): outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) A_d = outputs['A_d'] A_i = outputs['A_i'] theta_d = outputs['theta_d'] theta_i = outputs['theta_i'] theta_m = 0.00000001 R_d = outputs['R_d'] R_i = outputs['R_i'] theta, phi = np.linspace(0,2* np.pi, 200), np.linspace(pi-theta_m, np.pi, 200) THETA, PHI = np.meshgrid(theta, phi) R = 0.1*A_t/np.sin(theta_m) X = R * np.sin(PHI) * np.cos(THETA) Y = R * np.sin(PHI) * np.sin(THETA) Z = R * np.cos(PHI) Z = Z-np.max(Z)+0 height = np.full_like(Z, H_i) spherecoords.append(np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T) spherecoords = np.concatenate(spherecoords) viewer.add_points(np.array(spherecoords), name='cell wall', size=20, face_color='red', edge_color = 'transparent', blending='additive', opacity=0.6)
<Points layer 'cell wall' at 0x7f3d216805b0>
- I don't understand why this doesn't really work
vertices = spherecoords #faces = np.array([[0, 0, 1, 2], [0, 1, 2, 3], [0, 2, 3, 4]]) faces_list = [] counter = 0 for H_i in range(len(H_i_range)): for point in range(38): pointa = counter + point faces_list.append([pointa, pointa+1, pointa+2]) counter += 40 faces = np.array(faces_list) values = np.linspace(0, 1, len(vertices)) surface = (vertices, faces, values) viewer.add_surface(surface, colormap='viridis') # add the surface
<Surface layer 'surface' at 0x7f9405b20130>
- surface
examples:
vertices = np.array([[0, 0, 0, 0], [0, 10, 0, 20], [0, 20, 10, 0], [0, 0, 10, 10], [1, 20, 0, 0], [1, 10, 0, 20], [1, 20, 10, 0], [1, 0, 10, 10]]) faces = np.array([[0, 1, 2], [1, 2, 3], [4,5,6],[5,6,7]]) values = np.linspace(0.9, 1, len(vertices)) surface = (vertices, faces, values) viewer = napari.view_surface(surface) # add the surface facesloop = [] for t in (0,1): vert = vertices[vertices[:,0] == t] facest = scipy.spatial.Delaunay(vert[:,2:]).simplices + len(vert)*t facesloop.append(facest) facesd = np.concatenate(facesloop) viewer = napari.view_surface((vertices, facesd, values))
coords = np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T[:,1:] faces = scipy.spatial.Delaunay( coords[:,1:]).simplices values = np.linspace(0,1,len(coords)) viewer = napari.view_surface((coords[:,1:], faces, values))
%gui qt viewer = napari.Viewer(ndisplay=3) spherecoords = [] faces = [] for H_i in range(1,101): outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) A_d = outputs['A_d'] A_i = outputs['A_i'] theta_d = outputs['theta_d'] theta_i = outputs['theta_i'] R_d = outputs['R_d'] R_i = outputs['R_i'] theta, phi = np.linspace(0,2* np.pi, 50), np.linspace(pi-theta_d, np.pi, 50) THETA, PHI = np.meshgrid(theta, phi) R = R_d X = R * np.sin(PHI) * np.cos(THETA) Y = R * np.sin(PHI) * np.sin(THETA) Z = R * np.cos(PHI) Z = Z-np.max(Z) height = np.full_like(Z, H_i) vertices = np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T spherecoords.append(vertices) #face = scipy.spatial.Delaunay(vertices[:,2:]).simplices + len(vertices)*(H_i-1) if H_i == 1: face = scipy.spatial.Delaunay(vertices[:,2:]).simplices + len(vertices)*(H_i-1) else: face = faces[-1] + len(vertices) faces.append(face) spherecoords = np.concatenate(spherecoords) faces = np.concatenate(faces) surface_values = np.full(len(spherecoords),1) viewer.add_surface((spherecoords,faces,surface_values), name='droplet', colormap='green', blending='translucent', shading='smooth') spherecoords = [] faces = [] for H_i in range(1,101): outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) A_d = outputs['A_d'] A_i = outputs['A_i'] theta_d = outputs['theta_d'] theta_i = outputs['theta_i'] R_d = outputs['R_d'] R_i = outputs['R_i'] theta, phi = np.linspace(0,2* np.pi, 50), np.linspace(pi-theta_i, np.pi, 50) THETA, PHI = np.meshgrid(theta, phi) R = R_i X = R * np.sin(PHI) * np.cos(THETA) Y = R * np.sin(PHI) * np.sin(THETA) Z = R * np.cos(PHI) Z = Z-np.max(Z) height = np.full_like(Z, H_i) vertices = np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T spherecoords.append(vertices) if H_i == 1: face = scipy.spatial.Delaunay(vertices[:,2:]).simplices + len(vertices)*(H_i-1) else: face = faces[-1] + len(vertices) faces.append(face) spherecoords = np.concatenate(spherecoords) faces = np.concatenate(faces) surface_values = np.full(len(spherecoords),1) viewer.add_surface((spherecoords,faces,surface_values), name='coat', colormap='bop blue', blending='additive', shading='smooth') spherecoords = [] faces = [] for H_i in range(1,101): outputs = llps_model(H_i=H_i, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) A_d = outputs['A_d'] A_i = outputs['A_i'] theta_d = outputs['theta_d'] theta_i = outputs['theta_i'] theta_m = 0.00000001 R_d = outputs['R_d'] R_i = outputs['R_i'] theta, phi = np.linspace(0,2* np.pi, 50), np.linspace(pi-theta_m, np.pi, 50) THETA, PHI = np.meshgrid(theta, phi) R = 0.1*A_t/np.sin(theta_m) X = R * np.sin(PHI) * np.cos(THETA) Y = R * np.sin(PHI) * np.sin(THETA) Z = R * np.cos(PHI) Z = Z-np.max(Z)+0 height = np.full_like(Z, H_i) vertices = np.array([np.ravel(height), np.ravel(Z), np.ravel(Y), np.ravel(X)]).T spherecoords.append(vertices) #face = scipy.spatial.Delaunay(vertices[:,2:]).simplices + len(vertices)*(H_i-1) if H_i == 1: face = scipy.spatial.Delaunay(vertices[:,2:]).simplices + len(vertices)*(H_i-1) else: face = faces[-1] + len(vertices) faces.append(face) spherecoords = np.concatenate(spherecoords) faces = np.concatenate(faces) surface_values = np.full(len(spherecoords),1) viewer.add_surface((spherecoords,faces,surface_values), name='cell wall', colormap='bop blue', blending='opaque', shading='smooth') def update_frame(): """Update frame or time""" viewer.text_overlay.text = 'h = %1.2f\nblue: membrane\ngreen: condensate' % ((viewer.dims.current_step[0]+1)/100) viewer.dims.events.current_step.connect(lambda event: update_frame()) viewer.text_overlay.visible=True viewer.text_overlay.font_size=20 viewer.text_overlay.color = 'white' viewer.scale_bar.unit = 'nm' viewer.scale_bar.visible=True viewer.scale_bar.font_size=10
nil:END:
animation script
from napari_animation import Animation animation = Animation(viewer) viewer.reset_view() viewer.dims.current_step = (0, 0,0,0) animation.capture_keyframe(steps=15) animation.capture_keyframe(steps=15) viewer.camera.zoom = 2.6 viewer.camera.angles=(-0.026467524771271397, 0.5719902891444765, 11.550928988495166) animation.capture_keyframe(steps=50) viewer.dims.current_step = (viewer.dims.range[0][1]-1, 0,0,0) animation.capture_keyframe(steps=int(viewer.dims.range[0][1]-1)) viewer.reset_view() animation.capture_keyframe(steps=50) viewer.dims.current_step = (0, 0,0,0) animation.capture_keyframe(steps=int(viewer.dims.range[0][1]-1)) animation.animate('~/google_drive/grad_school/db_lab/code/analysis/llps_endocytosis_modeling/scripted_surface_animation.mov')
:results:
IMAGEIO FFMPEG_WRITER WARNING: input image is not divisible by macro_block_size=16, resizing from (2292, 1318) to (2304, 1328) to ensure video compatibility with most codecs and players. To prevent resizing, make your input image divisible by the macro_block_size or set the macro_block_size to 1 (risking incompatibility). 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- probably don't need this stuff
num_plots = 1 outputs = llps_model(H_i=90, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P) A_d = outputs['A_d'] A_i = outputs['A_i'] theta_d = outputs['theta_d'] theta_i = outputs['theta_i'] R_d = outputs['R_d'] R_i = outputs['R_i'] #fig, ax = plt.subplots(1,num_plots) #plt.subplot(1, num_plots, 1) #fig = plt.figure() #fig.set_size_inches((fig_width,fig_height)) #ax = fig.add_subplot(1, num_plots, 1) #plt.hlines([0,0],[-A_t,A_d],[-A_d,A_t], color='r', label='$S_o$') #plt.hlines([0,0],[-A_d,A_i],[-A_i,A_d], color='g', label='$S_m$') if V_d > 0: points_d = np.linspace(pi/2-theta_d, theta_d+pi/2, 1000) x_d = R_d*np.cos(points_d) y_d = R_d*np.sin(points_d) y_d_adj = y_d-np.min(y_d) points_i = np.linspace(pi/2-theta_i, theta_i+pi/2, 1000) x_i = R_i*np.cos(points_i) y_i = R_i*np.sin(points_i) y_i_adj = y_i-np.min(y_i) xy_i = np.array([y_i,x_i]).T viewer = napari.Viewer() viewer.add_points(xy_i)
<Points layer 'xy_i' at 0x7f9405887370>
nx, ny, nz = (3, 3, 4) x = np.linspace(0, 10, nx) y = np.linspace(0, 10, ny) z = np.linspace(0, 10, nz) xv, yv, zv = np.meshgrid(x, y, z) zyxv = np.array([np.ravel(zv), np.ravel(yv), np.ravel(xv)]).T zyxv viewer = napari.Viewer() viewer.add_points(zyxv)
<Points layer 'zyxv' at 0x7f93f2b41fd0>
import numpy as np import matplotlib.pyplot as plt from mpl_toolkits import mplot3d ax = plt.axes(projection="3d") xlist=np.linspace(-1.0,1.0,10)*100 ylist=np.linspace(-1.0,1.0,10)*100 r=np.linspace(1.0,1.0,10)*100 X,Y= np.meshgrid(xlist,ylist) Z=np.sqrt(r**2-X**2-Y**2) #Use np.sqrt like you had before spherecoords = np.array([np.ravel(Z), np.ravel(Y), np.ravel(X)]).T viewer.add_points(spherecoords) vertices = spherecoords faces = np.array([[0, 50, 99], [1, 2, 3]]) values = np.linspace(1, 1, len(vertices)) surface = (vertices, faces, values) viewer.add_surface(surface) cp=ax.plot_wireframe(X,Y,Z,color="r") cp=ax.plot_wireframe(X,Y,-Z,color="r") # Now plot the bottom half plt.title('2D Contour Plot of The unit sphere') plt.show()
<ipython-input-21-c34f0052e1a6>:11: RuntimeWarning: invalid value encountered in sqrt Z=np.sqrt(r**2-X**2-Y**2) #Use np.sqrt like you had before
- side-by-side delta plot and equilibrium geometry
def plot_simulation_geometry(x=H_i_range/vesicle_diameter, output='F', H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P, fig_width=15, fig_height=5, label=None, variable_df=variable_df, num_plots=2, top=20, bottom=-250, left=-500, right=500): outputs = llps_model(H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d) min_energy_ix = np.argmin(outputs['F']) H_i_eq = H_i_range[min_energy_ix] fig = show_geometry(H_i=H_i_eq, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P, fig_width=fig_width, fig_height=fig_height, num_plots=num_plots, top=top, bottom=bottom, left=left, right=right) plt.title('geometry at equilibrium') ax2 = fig.add_subplot(1,2,2) #y = outputs[output] #plt.plot(x,y,label=label) #plt.ylabel(output) plot_simulation_delta(x=x, output=output, H_i=H_i_range, sigma=sigma, A_t=A_t, S_i=S_i, K=K, C_i=C_i, gamma_d=gamma_d, gamma_m=gamma_m, gamma_i=gamma_i, V_d=V_d, P=P, label=label, variable_df=variable_df) plt.tight_layout()
:results: nil:END:
- side-by-side F plots and parameter phase diagram
def boundary_test(x_param, y_param, x_range, boundary, params=params, factors=[0.7,1,1.3], y_offset=0.2, x_scale='log'): fparams = params.copy() fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize=(10,4)) if x_scale == 'log': ax1.semilogx(x_range, boundary, color='black') else: ax1.plot(x_range, boundary, color='black') for factor in factors: middle = len(x_range)//2 x_pick = x_range[middle]*factor boundary_pick = boundary[middle]+(factor-1)*np.sqrt(boundary[middle]**2)*y_offset fparams[x_param] = x_pick fparams[y_param] = boundary_pick F = llps_model_recbrt(h=fparams['h_range'], sigma=fparams['sigma'], A_t=fparams['A_t'], S_i=fparams['S_i'], K=fparams['K'], C_i=fparams['C_i'], gamma_d=fparams['gamma_d'], gamma_m=fparams['gamma_m'], gamma_i=fparams['gamma_i'], V_d=fparams['V_d'], P=fparams['P']) delta_F = F-F[0] ax1.scatter(x_pick, boundary_pick) ax2.plot(h_range,delta_F) ax2.hlines(0, 0,1.05, color='black') ax1.set(xlabel=x_param, ylabel=y_param) ax2.set(xlabel='h', ylabel='∆F') plt.tight_layout()
:results: nil:END:
- global stability parameter analysis
[4/4]
- DONE \(K\)
from therm_spont_k
def thermo_spont_K(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d, V_d=V_d, S_i=S_i, P=P, C_i=C_i): alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) eta = (S_i**(3/2))/(6*pi**(1/2)) K = ((-(sigma*alpha+(sigma-gamma_m)*beta+gamma_d*epsilon)* (np.cbrt(3*V_d+(S_i**(3/2))/(2*pi**(1/2)))**2- np.cbrt(3*V_d)**2)+2*P*eta-(sigma-gamma_m)*S_i- 3*P*eta)/(8*pi-8*C_i*(pi*S_i)**(1/2))) return K
:results: nil:END:
- testing
V_d_range = np.logspace(6,9) K_boundary = thermo_spont_K(V_d=V_d_range) boundary_test(x_param='V_d', y_param='K', x_range=V_d_range, boundary=K_boundary)
C_i_range = np.logspace(-4,-2.1,1000) K_boundary = thermo_spont_K(C_i=C_i_range) boundary_test(x_param='C_i', y_param='K', x_range=C_i_range, boundary=K_boundary, y_offset=-0.3)
- this kind of makes sense - the lower the spontaneous curvature (flatter), the more the energetics are sensitive to rigidity
x_range = np.logspace(6,9) thermo_boundary = thermo_spont_K(V_d=x_range) kinetic_boundary = kinetic_spont_K(V_d=x_range) both_boundary_test(x_param='V_d', y_param='K', x_range=x_range, thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, y_offset=2)
--------------------------------------------------------------------------- NameError Traceback (most recent call last) Cell In [24], line 3 1 x_range = np.logspace(6,9) 2 thermo_boundary = thermo_spont_K(V_d=x_range) ----> 3 kinetic_boundary = kinetic_spont_K(V_d=x_range) 4 both_boundary_test(x_param='V_d', y_param='K', x_range=x_range, 5 thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, 6 y_offset=2) NameError: name 'kinetic_spont_K' is not defined
- testing
- DONE \(P\)
- from therm_spont_p
def thermo_spont_P(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d, V_d=V_d, S_i=S_i, K=K, C_i=C_i): alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) eta = (S_i**(3/2))/(6*pi**(1/2)) P = ((sigma*alpha + (sigma - gamma_m)*beta + gamma_d*epsilon)*(np.cbrt(3*V_d)**2 - np.cbrt(S_i**(3/2)/(2*np.sqrt(pi)) + 3*V_d)**2) - (sigma-gamma_m)*S_i - 8*K*(pi - C_i*np.sqrt(pi*S_i)))/ eta return P
:results: nil:END:
- DONE \(\sigma\)
- from therm_spont_sigma
def thermo_spont_sigma(P=P, gamma_m=gamma_m, gamma_d=gamma_d, V_d=V_d, S_i=S_i, K=K, C_i=C_i): alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) eta = (S_i**(3/2))/(6*pi**(1/2)) sigma = ((-gamma_m*S_i+8*K*(pi-C_i*np.sqrt(pi*S_i))+P*eta)/ ((3*V_d)**(2/3)-((S_i**(3/2))/ (2*np.sqrt(pi))+3*V_d)**(2/3))- gamma_d*epsilon+gamma_m*beta)/(alpha+beta- (S_i)/((3*V_d)**(2/3)-((S_i**(3/2))/ (2*np.sqrt(pi))+3*V_d)**(2/3))) return sigma
:results: nil:END:
- DONE \(C_{i}\)
- from therm_spont_ci
def thermo_spont_C_i(P=P, gamma_m=gamma_m, gamma_d=gamma_d, V_d=V_d, S_i=S_i, K=K, sigma=sigma): alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) eta = (S_i**(3/2))/(6*pi**(1/2)) C_i = (pi-((sigma*alpha+(sigma-gamma_m)*beta+gamma_d*epsilon)*(np.cbrt(3*V_d)**2-np.cbrt((S_i**(3/2))/(2*np.sqrt(pi))+3*V_d)**2)-(sigma-gamma_m)*S_i-P*eta)/(8*K))/(np.sqrt(pi*S_i)) return C_i
:results: nil:END:
- DONE \(K\)
- local stability parameter analysis
[8/8]
- DONE \(K\)
from kinetic_spont_k
def kinetic_spont_K(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d, V_d=V_d, S_i=S_i, P=P, C_i=C_i): alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) eta = (S_i**(3/2))/(6*pi**(1/2)) K = (((sigma*alpha+(sigma-gamma_m)*beta+gamma_d*epsilon)*S_i**(3/2))/(np.sqrt(pi)*np.cbrt(3*V_d)) + 3*P*eta)/(8*C_i*np.sqrt(pi*S_i)) return K
:results: nil:END:
- DONE \(P\)
from kinetic_spont_p
def kinetic_spont_P(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d, V_d=V_d, S_i=S_i, K=K, C_i=C_i): alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) eta = (S_i**(3/2))/(6*pi**(1/2)) P = (-((sigma*alpha + (sigma - gamma_m)*beta +gamma_d*epsilon)*S_i**(3/2))/ (np.sqrt(pi)*np.cbrt(3*V_d)) + 8*K*C_i*np.sqrt(pi*S_i))/(3*eta) return P
:results: nil:END:
- DONE \(\sigma\)
nil:END: :results: nil:END: :results: nil:END:
:RESULTS:
--------------------------------------------------------------------------- NameError Traceback (most recent call last) <ipython-input-35-594c148ecf9a> in <module> ----> 1 def kinetic_spont_sigma(P=P, gamma_m=gamma_m, gamma_d=gamma_d, 2 V_d=V_d, S_i=S_i, K=K, C_i=C_i): 3 4 alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) 5 beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) NameError: name 'P' is not defined- testing
x_range = np.logspace(6,9) boundary = kinetic_spont_sigma(V_d=x_range) boundary_test(x_param='V_d', y_param='sigma', x_range=x_range, boundary=boundary, y_offset=-0.5)
--------------------------------------------------------------------------- TypeError Traceback (most recent call last) Cell In [36], line 2 1 x_range = np.logspace(6,9) ----> 2 boundary = kinetic_spont_sigma(V_d=x_range) 3 boundary_test(x_param='V_d', y_param='sigma', x_range=x_range, boundary=boundary, 4 y_offset=-0.5) Cell In [35], line 9, in kinetic_spont_sigma(P, gamma_m, gamma_d, V_d, S_i, K, C_i) 6 epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) 7 eta = (S_i**(3/2))/(6*pi**(1/2)) ----> 9 sigma = (( np.sqrt(pi)/S_i**(3/2) )*np.cbrt(3*V_d)(8*K*C_i*np.sqrt(pi*S_i)-3*P*eta)- 10 gamma_d*epsilon+gamma_m*beta)/(alpha+beta) 12 return sigma TypeError: 'numpy.ndarray' object is not callable:END:
- testing
- DONE \(C_{i}\)
from kinetic_spont_ci
def kinetic_spont_C_i(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d, V_d=V_d, S_i=S_i, K=K, P=P): alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) eta = (S_i**(3/2))/(6*pi**(1/2)) C_i = (((sigma*alpha + (sigma - gamma_m)*beta +gamma_d*epsilon)*S_i**(3/2))/ (np.sqrt(pi)*np.cbrt(3*V_d)) + 3*P*eta)/(8*K*np.sqrt(pi*S_i)) return C_i
:results: nil:END:
- CANCELLED \(\gamma_m\)
- CANCELLED \(\gamma_d\)
- CANCELLED \(S_{i}\)
- DONE \(V_{d}\)
from kinetic_spont_ci
def kinetic_spont_V_d(sigma=sigma, gamma_m=gamma_m, gamma_d=gamma_d, C_i=C_i, S_i=S_i, K=K, P=P): alpha = np.cbrt((pi*(gamma_m**2-gamma_d**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) beta = np.cbrt((pi*(gamma_d**2-gamma_m**2)**3)/((2*gamma_d+gamma_m)**2*(gamma_d-gamma_m)**4)) epsilon = np.cbrt((8*pi)/((2+gamma_m/gamma_d)**2*(1-gamma_m/gamma_d))) eta = (S_i**(3/2))/(6*pi**(1/2)) V_d = (1/3)*(((sigma*alpha + (sigma - gamma_m)*beta +gamma_d*epsilon)*S_i**(3/2))/ ((8*K*C_i*np.sqrt(pi*S_i)-3*P*eta)*np.sqrt(pi)))**3 return V_d
:results: nil:END:
- DONE \(K\)
- side-by-side F plots with both phase boundaries
def both_boundary_test(x_param, y_param, x_range, thermo_boundary, kinetic_boundary, params=params, factors=[0.7,1,1.3], y_offset=0.2, x_scale='log', y_scale='linear', variable_df=variable_df): fparams = params.copy() fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize=(10,4)) # if x_scale == 'log': # ax1.semilogx(x_range, thermo_boundary, color='black', label='thermodynamic') # ax1.semilogx(x_range, kinetic_boundary, color='blue', label='kinetic') # # else: # ax1.plot(x_range, boundary, color='black') if y_offset > 0: fill_range = np.max(kinetic_boundary) if y_offset < 0: fill_range = np.min(kinetic_boundary) ax1.loglog(x_range, thermo_boundary, color='yellow') ax1.fill_between(x_range, y1=thermo_boundary, y2=fill_range*1.3, color='yellow', alpha=0.8, label='globally stable') ax1.loglog(x_range, kinetic_boundary, color='blue') ax1.fill_between(x_range, y1=kinetic_boundary, y2=fill_range*1.3, color='blue', alpha=0.8, label='locally stable') ax1.legend() for i, factor in enumerate(factors): middle = len(x_range)//2 x_pick = x_range[middle]*factor for j, boundary in enumerate([thermo_boundary, kinetic_boundary]): boundary_pick = boundary[middle]+(factor-1)*np.sqrt(thermo_boundary[middle]**2)*y_offset fparams[x_param] = x_pick fparams[y_param] = boundary_pick F = llps_model_recbrt(h=fparams['h_range'], sigma=fparams['sigma'], A_t=fparams['A_t'], S_i=fparams['S_i'], K=fparams['K'], C_i=fparams['C_i'], gamma_d=fparams['gamma_d'], gamma_m=fparams['gamma_m'], gamma_i=fparams['gamma_i'], V_d=fparams['V_d'], P=fparams['P']) delta_F = F-F[0] # ax1.scatter(x_pick, boundary_pick, linewidths=5, color=((j)/2, (j)/2, (j+1)/2, (i+1)/len(factors))) # ax2.plot(h_range,delta_F, color=((j)/2, (j)/2, (j+1)/2, (i+1)/len(factors))) ax1.scatter(x_pick, boundary_pick, linewidths=5) ax2.plot(h_range,delta_F) ax2.hlines(0, -0.05,1.05, color='black') ax1.set(xlabel=variable_df.loc[x_param]['description']+' ('+variable_df.loc[x_param]['units']+')', ylabel=variable_df.loc[y_param]['description']+' ('+variable_df.loc[y_param]['units']+')') # xscale=x_scale) ax1.set_xscale(x_scale) ax1.set_yscale(y_scale) ax2.set(xlabel='h: relative invagination height', ylabel='∆F ($pN \cdot nm$)') plt.tight_layout()
:results: nil:END:
- model functions
3.2.2. result plots [8/9]
- plot all outputs vs. invagination height for default parameters
- as individual plots
outputs = llps_model() outputs.pop('theta_d') # outputs.pop('2÷R_i') variables = list(outputs.keys()) for variable in variables: plt.figure() plot_simulation(output=variable) plt.xlabel('h: invagination distance/vesicle diameter') plt.savefig(fig_dir+'%s-vs-H_i.png' %(variable))
--------------------------------------------------------------------------- KeyError Traceback (most recent call last) File ~/.local/lib/python3.10/site-packages/pandas/core/indexes/base.py:3803, in Index.get_loc(self, key, method, tolerance) 3802 try: -> 3803 return self._engine.get_loc(casted_key) 3804 except KeyError as err: File ~/.local/lib/python3.10/site-packages/pandas/_libs/index.pyx:138, in pandas._libs.index.IndexEngine.get_loc() File ~/.local/lib/python3.10/site-packages/pandas/_libs/index.pyx:165, in pandas._libs.index.IndexEngine.get_loc() File pandas/_libs/hashtable_class_helper.pxi:5745, in pandas._libs.hashtable.PyObjectHashTable.get_item() File pandas/_libs/hashtable_class_helper.pxi:5753, in pandas._libs.hashtable.PyObjectHashTable.get_item() KeyError: 'bending_energy_flipped' The above exception was the direct cause of the following exception: KeyError Traceback (most recent call last) Cell In [42], line 8 6 for variable in variables: 7 plt.figure() ----> 8 plot_simulation(output=variable) 9 plt.xlabel('h: invagination distance/vesicle diameter') 10 plt.savefig(fig_dir+'%s-vs-H_i.png' %(variable)) Cell In [12], line 9, in plot_simulation(x, output, H_i, sigma, A_t, S_i, K, C_i, gamma_d, gamma_m, gamma_i, V_d, P, label, variable_df) 7 y = outputs[output] 8 plt.plot(x,y,label=label) ----> 9 plt.ylabel(variable_df.loc[output]['description']+' ('+variable_df.loc[output]['units']+')') File ~/.local/lib/python3.10/site-packages/pandas/core/indexing.py:1073, in _LocationIndexer.__getitem__(self, key) 1070 axis = self.axis or 0 1072 maybe_callable = com.apply_if_callable(key, self.obj) -> 1073 return self._getitem_axis(maybe_callable, axis=axis) File ~/.local/lib/python3.10/site-packages/pandas/core/indexing.py:1312, in _LocIndexer._getitem_axis(self, key, axis) 1310 # fall thru to straight lookup 1311 self._validate_key(key, axis) -> 1312 return self._get_label(key, axis=axis) File ~/.local/lib/python3.10/site-packages/pandas/core/indexing.py:1260, in _LocIndexer._get_label(self, label, axis) 1258 def _get_label(self, label, axis: int): 1259 # GH#5567 this will fail if the label is not present in the axis. -> 1260 return self.obj.xs(label, axis=axis) File ~/.local/lib/python3.10/site-packages/pandas/core/generic.py:4056, in NDFrame.xs(self, key, axis, level, drop_level) 4054 new_index = index[loc] 4055 else: -> 4056 loc = index.get_loc(key) 4058 if isinstance(loc, np.ndarray): 4059 if loc.dtype == np.bool_: File ~/.local/lib/python3.10/site-packages/pandas/core/indexes/base.py:3805, in Index.get_loc(self, key, method, tolerance) 3803 return self._engine.get_loc(casted_key) 3804 except KeyError as err: -> 3805 raise KeyError(key) from err 3806 except TypeError: 3807 # If we have a listlike key, _check_indexing_error will raise 3808 # InvalidIndexError. Otherwise we fall through and re-raise 3809 # the TypeError. 3810 self._check_indexing_error(key) KeyError: 'bending_energy_flipped'
- as subplots of one figure
outputs = llps_model() outputs.pop('theta_d') variables = list(outputs.keys()) nplots = len(variables) width = 3 height = nplots//3+1 plt.figure(figsize=(width*5,height*4)) for i, variable in enumerate(variables): plt.subplot(height, width, i+1) plot_simulation(output=variable) plt.xlabel('h: invagination distance/vesicle diameter') plt.tight_layout() plt.savefig(fig_dir+'all_outputs.png')
--------------------------------------------------------------------------- KeyError Traceback (most recent call last) File ~/.local/lib/python3.10/site-packages/pandas/core/indexes/base.py:3803, in Index.get_loc(self, key, method, tolerance) 3802 try: -> 3803 return self._engine.get_loc(casted_key) 3804 except KeyError as err: File ~/.local/lib/python3.10/site-packages/pandas/_libs/index.pyx:138, in pandas._libs.index.IndexEngine.get_loc() File ~/.local/lib/python3.10/site-packages/pandas/_libs/index.pyx:165, in pandas._libs.index.IndexEngine.get_loc() File pandas/_libs/hashtable_class_helper.pxi:5745, in pandas._libs.hashtable.PyObjectHashTable.get_item() File pandas/_libs/hashtable_class_helper.pxi:5753, in pandas._libs.hashtable.PyObjectHashTable.get_item() KeyError: 'bending_energy_flipped' The above exception was the direct cause of the following exception: KeyError Traceback (most recent call last) Cell In [43], line 11 9 for i, variable in enumerate(variables): 10 plt.subplot(height, width, i+1) ---> 11 plot_simulation(output=variable) 12 plt.xlabel('h: invagination distance/vesicle diameter') 14 plt.tight_layout() Cell In [12], line 9, in plot_simulation(x, output, H_i, sigma, A_t, S_i, K, C_i, gamma_d, gamma_m, gamma_i, V_d, P, label, variable_df) 7 y = outputs[output] 8 plt.plot(x,y,label=label) ----> 9 plt.ylabel(variable_df.loc[output]['description']+' ('+variable_df.loc[output]['units']+')') File ~/.local/lib/python3.10/site-packages/pandas/core/indexing.py:1073, in _LocationIndexer.__getitem__(self, key) 1070 axis = self.axis or 0 1072 maybe_callable = com.apply_if_callable(key, self.obj) -> 1073 return self._getitem_axis(maybe_callable, axis=axis) File ~/.local/lib/python3.10/site-packages/pandas/core/indexing.py:1312, in _LocIndexer._getitem_axis(self, key, axis) 1310 # fall thru to straight lookup 1311 self._validate_key(key, axis) -> 1312 return self._get_label(key, axis=axis) File ~/.local/lib/python3.10/site-packages/pandas/core/indexing.py:1260, in _LocIndexer._get_label(self, label, axis) 1258 def _get_label(self, label, axis: int): 1259 # GH#5567 this will fail if the label is not present in the axis. -> 1260 return self.obj.xs(label, axis=axis) File ~/.local/lib/python3.10/site-packages/pandas/core/generic.py:4056, in NDFrame.xs(self, key, axis, level, drop_level) 4054 new_index = index[loc] 4055 else: -> 4056 loc = index.get_loc(key) 4058 if isinstance(loc, np.ndarray): 4059 if loc.dtype == np.bool_: File ~/.local/lib/python3.10/site-packages/pandas/core/indexes/base.py:3805, in Index.get_loc(self, key, method, tolerance) 3803 return self._engine.get_loc(casted_key) 3804 except KeyError as err: -> 3805 raise KeyError(key) from err 3806 except TypeError: 3807 # If we have a listlike key, _check_indexing_error will raise 3808 # InvalidIndexError. Otherwise we fall through and re-raise 3809 # the TypeError. 3810 self._check_indexing_error(key) KeyError: 'bending_energy_flipped'
- as individual plots
- change in all outputs vs. invagination height for default parameters
- as individual plots
outputs = llps_model() outputs.pop('theta_d') variables = outputs.keys() for variable in variables: plt.figure() plot_simulation_delta(output=variable) plt.tight_layout() plt.savefig(fig_dir+'%s-vs-H_i_delta.png' %(variable))
- as subplots of one figure
outputs = llps_model() outputs.pop('theta_d') variables = outputs.keys() nplots = len(variables) width = 3 height = nplots//3+1 plt.figure(figsize=(width*5,height*4)) for i, variable in enumerate(variables): plt.subplot(height, width, i+1) plot_simulation_delta(output=variable) plt.tight_layout() plt.savefig(fig_dir+'all_outputs_delta.png')
- as individual plots
- change in surface areas
variables = ['S_o', 'S_d', 'S_m'] for variable in variables: plot_simulation_delta(output=variable, label=variable) plt.ylabel('∆ surface area') plt.legend() plt.tight_layout() plt.savefig(fig_dir+'areas-vs-H_i_delta.png')
- each individual energy term output
- set temporary variables
P_temp = 1e-3 gamma_d_temp = gamma_d*10 gamma_m_temp = gamma_m*10 K_temp = 160
:results: nil:END:
- normal droplet
show_geometry(H_i=100) plt.savefig(fig_dir+'normal_droplet.png')
- with appropriate signs
droplet only:
variables = ['surface_tension', 'wetting_energy', 'droplet_energy'] colors = ['blue', 'green', 'black'] plt.figure(figsize=(6,6)) for variable, color in zip(variables, colors): plot_simulation_delta(output=variable, label=variable, P=P_temp, gamma_d=gamma_d_temp, gamma_m=gamma_m_temp, K=K_temp, color=color) plt.legend() #plt.ylim(-100, 100) plt.ylabel('∆ energy (pN * nm)') #plt.title(title) plt.tight_layout() plt.savefig(fig_dir+'droplet_energy_contributions.png') plt.savefig(fig_dir+'droplet_energy_contributions.svg')
all contributions:
variables = ['surface_tension', 'wetting_energy', 'droplet_energy', 'membrane_tension', 'bending_energy', 'turgor_pressure', 'F'] colors = ['blue', 'green', 'black', 'red', 'purple', 'brown', 'orange'] plt.figure(figsize=(6,6)) for variable, color in zip(variables, colors): plot_simulation_delta(output=variable, label=variable, P=P_temp, gamma_d=gamma_d_temp, gamma_m=gamma_m_temp, K=50, color=color) plt.legend() #plt.ylim(-100, 100) plt.ylabel('∆ energy (pN * nm)') #plt.title(title) plt.tight_layout() plt.savefig(fig_dir+'energy_contributions.png') plt.savefig(fig_dir+'energy_contributions.svg')
- flipped sign of negative energy contributions
variables = ['membrane_tension', 'bending_energy_flipped','surface_tension', 'wetting_energy_flipped', 'turgor_pressure'] plt.figure(figsize=(10,10)) for variable in variables: plot_simulation_delta(output=variable, label=variable, P=P_temp, gamma_d=gamma_d_temp, gamma_m=gamma_m_temp, K=K_temp) plt.legend() #plt.ylim(-100, 100) plt.ylabel('∆ energy (pN * nm)') #plt.title(title) plt.tight_layout() plt.savefig(fig_dir+'energy_contributions_flipped.png')
- with appropriate signs
- huge droplet
show_geometry(H_i=100, V_d=V_d*10**4, left=-3600, right=3600, bottom=-2300) plt.savefig(fig_dir+'bigdrop.png')
- with appropriate signs
variables = ['surface_tension', 'wetting_energy', 'droplet_energy'] colors = ['blue', 'green', 'black'] plt.figure(figsize=(6,6)) for variable, color in zip(variables, colors): plot_simulation_delta(output=variable, label=variable, P=P_temp, gamma_d=gamma_d_temp, gamma_m=gamma_m_temp, K=K_temp, V_d=V_d*10**4, color=color) plt.legend() #plt.ylim(-100, 100) plt.ylabel('∆ energy (pN * nm)') #plt.title(title) plt.tight_layout() plt.savefig(fig_dir+'droplet_energy_contributions_bigdrop.png') plt.savefig(fig_dir+'droplet_energy_contributions_bigdrop.svg')
- flipped sign of negative energy contributions
variables = ['membrane_tension', 'bending_energy_flipped','surface_tension', 'wetting_energy_flipped', 'turgor_pressure'] plt.figure(figsize=(10,10)) for variable in variables: plot_simulation_delta(output=variable, label=variable, P=P_temp, gamma_d=gamma_d_temp, gamma_m=gamma_m_temp, K=K_temp, V_d=V_d*10**4) plt.legend() #plt.ylim(-100, 100) plt.ylabel('∆ energy (pN * nm)') #plt.title(title) plt.tight_layout() plt.savefig(fig_dir+'energy_contributions_bigdrop_flipped.png')
- with appropriate signs
- set temporary variables
- Plot F for a series of values of gammam
What are the parameter values? I assume that gammam is negative. Plot F for a series of values of gammam. Based on the plots for Sm, Sd, and S0, F should end up with a minimum at H/R=1 if gammam is large enough. It might also happen if gammad is small enough. It's hard to say because I can't easily compare the magnitude of changes in Sd, Si, and S0.
- absolute
gamma_m_range = [0,50,100,150,199] for i in gamma_m_range: plot_simulation(gamma_m=i, label=i/gamma_d) plt.legend(title='gamma_m/gamma_d') plt.xlabel('invagination distance/vesicle diameter') plt.tight_layout() plt.savefig(fig_dir+'f_scan_gamma_m.png')
/tmp/ipykernel_195576/574978439.py:6: RuntimeWarning: invalid value encountered in arccos theta_d = np.arccos(gamma_m/gamma_d) #equation 8
- normalized
gamma_m_range = [0,50,100,150,199] for i in gamma_m_range: plot_simulation_normalized(gamma_m=i, label=i/gamma_d) plt.legend(title='gamma_m/gamma_d') plt.xlabel('invagination distance/vesicle diameter') plt.tight_layout() plt.savefig(fig_dir+'f_scan_gamma_m.png')
/tmp/ipykernel_195576/574978439.py:6: RuntimeWarning: invalid value encountered in arccos theta_d = np.arccos(gamma_m/gamma_d) #equation 8
:end:
- delta
gamma_m_range = np.array( [0.01,0.25,0.5,0.75] )*gamma_d for i in gamma_m_range: plot_simulation_delta(gamma_m=i, label=i/gamma_d) plt.legend(title='$\gamma_{m}/\gamma_{d}$') plt.tight_layout() plt.savefig(fig_dir+'f_scan_gamma_m_delta.png')
- delta and geometry side by side
gamma_m_range = [0,50,100,150,180] for i in gamma_m_range: plot_simulation_geometry(gamma_m=i, label=i/gamma_d) plt.legend(title='$\gamma_{m}/\gamma_{d}$') plt.tight_layout() plt.savefig(fig_dir+'f_scan_gamma_m_%s.png' %(i))
/tmp/ipykernel_195576/574978439.py:6: RuntimeWarning: invalid value encountered in arccos theta_d = np.arccos(gamma_m/gamma_d) #equation 8 /home/maxferrin/miniconda3/envs/llps_model_analysis/lib/python3.10/site-packages/matplotlib/axes/_base.py:2480: UserWarning: Warning: converting a masked element to nan. xys = np.asarray(xys)
- absolute
- F over series of turgor pressure
pressure_range = np.linspace(10**-2,1,5) for i in pressure_range: plot_simulation_delta(P=i, label=i) plt.legend(title='turgor pressure ($N/nm^{2}$)') plt.tight_layout() plt.savefig(fig_dir+'f_scan_turgor_delta.png')
- spontaneity analysis
there's no solution for kinetic \(\sigma\)
- DONE \(V_{d}\) x-axis
boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2), (thermo_spont_P,kinetic_spont_P, 'P', -2), #(thermo_spont_sigma,kinetic_spont_sigma, 'sigma'), (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)] x_range = np.logspace(7,9,base=10) for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries: thermo_boundary = thermo_boundaryy(V_d=x_range) kinetic_boundary = kinetic_boundaryy(V_d=x_range) both_boundary_test(x_param='V_d', y_param=y_param, x_range=x_range, thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, y_offset=y_offset/2, factors=[0.9,1,1.1]) plt.savefig(fig_dir+y_param+'V_d_boundary_check.png')
show_geometry(H_i=100, V_d=10**(7)) plt.savefig(fig_dir+'10e7_geometry.png')
show_geometry(H_i=100, V_d=10**(9), left=-1200, right=1200, bottom=-600) plt.savefig(fig_dir+'10e9_geometry.png')
- DONE \(\gamma_{d}\) x-axis
boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2), (thermo_spont_P,kinetic_spont_P, 'P', -2), #(thermo_spont_sigma,kinetic_spont_sigma, 'sigma'), (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)] x_range = np.logspace(-4,-1,base=10) for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries: thermo_boundary = thermo_boundaryy(gamma_d=x_range) kinetic_boundary = kinetic_boundaryy(gamma_d=x_range) both_boundary_test(x_param='gamma_d', y_param=y_param, x_range=x_range, thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, y_offset=y_offset/2, factors=[0.9,1,1.1]) plt.savefig(fig_dir+y_param+'gamma_d_boundary_check.png')
- DONE \(\gamma_{m}\) x-axis
boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2), (thermo_spont_P,kinetic_spont_P, 'P', -2), #(thermo_spont_sigma,kinetic_spont_sigma, 'sigma'), (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)] x_range = np.logspace(-4,-1,base=10) for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries: thermo_boundary = thermo_boundaryy(gamma_m=x_range) kinetic_boundary = kinetic_boundaryy(gamma_m=x_range) both_boundary_test(x_param='gamma_m', y_param=y_param, x_range=x_range, thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, y_offset=y_offset*2, factors=[0.8,1,1.2]) plt.savefig(fig_dir+y_param+'gamma_m_boundary_check.png')
- TODO \(\gamma_{m}\) /\(\gamma_{d}\) x-axis
boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2), (thermo_spont_P,kinetic_spont_P, 'P', -2), #(thermo_spont_sigma,kinetic_spont_sigma, 'sigma'), (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)] x_range = np.logspace(-2,0.0001,base=10)*gamma_d for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries: thermo_boundary = thermo_boundaryy(gamma_m=x_range) kinetic_boundary = kinetic_boundaryy(gamma_m=x_range) both_boundary_test(x_param='gamma_m', y_param=y_param, x_range=x_range, thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, y_offset=y_offset/2, factors=[0.8,1,1.2]) plt.savefig(fig_dir+y_param+'gamma_m_relative_boundary_check.png')
- DONE \(\sigma\) x-axis
boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', -2), (thermo_spont_P,kinetic_spont_P, 'P', 2), (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', -2)] x_range = np.logspace(-5,-1,base=10) for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries: thermo_boundary = thermo_boundaryy(sigma=x_range) kinetic_boundary = kinetic_boundaryy(sigma=x_range) both_boundary_test(x_param='sigma', y_param=y_param, x_range=x_range, thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, y_offset=y_offset/5, factors=[0.8,1,1.2]) plt.savefig(fig_dir+y_param+'sigma_boundary_check.png')
- DONE \(S_{i}\) x-axis
boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2), (thermo_spont_P,kinetic_spont_P, 'P', -2), (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)] x_range = np.logspace(3.5,4.5,base=10) for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries: thermo_boundary = thermo_boundaryy(S_i=x_range) kinetic_boundary = kinetic_boundaryy(S_i=x_range) both_boundary_test(x_param='S_i', y_param=y_param, x_range=x_range, thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, y_offset=y_offset*50, factors=[0.999,1,1.001]) plt.savefig(fig_dir+y_param+'S_i_boundary_check.png')
- DONE \(P\) x-axis
boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2), #(thermo_spont_P,kinetic_spont_P, 'P', 2), (thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2)] x_range = np.logspace(-3,0,base=10) for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries: thermo_boundary = thermo_boundaryy(P=x_range) kinetic_boundary = kinetic_boundaryy(P=x_range) both_boundary_test(x_param='P', y_param=y_param, x_range=x_range, thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, y_offset=y_offset*150, factors=[0.999,1,1.001]) plt.savefig(fig_dir+y_param+'P_boundary_check.png')
- DONE \(C_{i}\) x-axis
boundaries = [(thermo_spont_K,kinetic_spont_K, 'K', 2), (thermo_spont_P,kinetic_spont_P, 'P', -2)] x_range = np.logspace(-4,-1,base=10) for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries: thermo_boundary = thermo_boundaryy(C_i=x_range) kinetic_boundary = kinetic_boundaryy(C_i=x_range) both_boundary_test(x_param='C_i', y_param=y_param, x_range=x_range, thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, y_offset=y_offset/5, factors=[0.8,1,1.2]) plt.savefig(fig_dir+y_param+'C_i_boundary_check.png')
- DONE \(K\) x-axis
boundaries = [(thermo_spont_C_i,kinetic_spont_C_i, 'C_i', 2), (thermo_spont_P,kinetic_spont_P, 'P', -2)] x_range = np.logspace(1,4,base=10) for thermo_boundaryy, kinetic_boundaryy, y_param, y_offset in boundaries: thermo_boundary = thermo_boundaryy(K=x_range) kinetic_boundary = kinetic_boundaryy(K=x_range) both_boundary_test(x_param='K', y_param=y_param, x_range=x_range, thermo_boundary=thermo_boundary, kinetic_boundary=kinetic_boundary, y_offset=y_offset/1, factors=[0.9,1,1.1]) plt.savefig(fig_dir+y_param+'K_boundary_check.png')
- DONE \(V_{d}\) x-axis
- plots for 20200915 lab meeting drafting
- TODO each individual energy term output
variables = ['membrane_tension', 'bending_energy','surface_tension', 'wetting_energy','turgor_pressure'] for variable in variables: plt.figure() plot_simulation_delta(output=variable) plt.tight_layout() plt.savefig(fig_dir+'%s-vs-H_i_delta.png' %(variable))
- membrane components without droplet
- droplet effects
- TODO droplet+membrane effects
- TODO turgor pressure effects
- TODO each individual energy term output
